How many three-digit numbers are there in the abacus dial five beads

Because it is a three-digit number,

[1] There must be beads in the hundreds place slot. [2] Beads can only be in the hundreds, tens, and single digits.

Also: it should be determined not to take into account whether the representation of the top row of 2 beads dialed down at the same time to actually count into one place is credited to the case.

Let's consider two of these cases: tens and digits have beads/tens or digits have one without beads.

For example: when there is 1 bead in the hundreds place, if there is 1 bead in the tens place and 3 beads in the ones place. At this time the number of digits that can be expressed in the first place **** 2, 3 (on the 0 under 3) or 7 (on the 1 under 2), the beads in the tenth place can be expressed in the number of digits ***2, 1 (on the 0 under 1) or 5 (on the 1 under 0), the beads in the hundredth place can be expressed in the number of digits **** 2, 1 (on the 0 under 1) or 5 (on the 1 under 0). Therefore, the ratio of the number of beads 1:1:3 *** there are 2x2x2*** 8 cases (representing 113, 117, 153, 157, 513, 517, 553, 557*** 8 numbers)

Another example: the hundredths place 1 bead, if the tens place has 0 beads, the first place has 4 beads. Hundreds of the same, ten zero can only indicate 1 number, the number of digits can be expressed **** 2, 4 (on the 0 under the 4) or 8 (on the 1 under the 3). Therefore, the ratio of the number of arithmetic beads for 1:0:4 **** have 2x1x2*** 4 kinds of cases (represent 104, 108, 504, 508*** 4 numbers)

Similarly, it can be seen:

The ratio of the number of arithmetic beads for 1:2:2, **** there are 2x2x2***8 kinds of cases; the ratio of the number of arithmetic beads for 1:3:1, **** there are 2x2x2**8 kinds of cases; arithmetic beads for 1:3:1, **** there are 2x2x2**8 kinds of cases; arithmetic beads for 1:3:1, **** there are 2x2x2 ***8 cases; when the ratio of the number of beads is 1:4:0, **** has 2x2x1***4 cases.

That is, when the hundredths digit is 1 bead, *** counts 4+8+8+8+4=32

After reasoning, we can see that when the hundredths digit is 2 beads, the ratio of beads is only 4 kinds of cases, and *** can represent 4+8+8+4=24 numbers; when the hundredths digit is 3 beads, the ratio of beads is only 3 kinds of cases, and *** can represent 4+8+4=16 numbers; when the when the hundredths digit is 4 beads, there are only 2 cases of arithmetic bead proportions, and *** can represent 4+4=8 numbers. Finally, special case, when the number of hundred digits is 5 beads, there is only 1 case of the bead ratio, *** can represent 1 number (900, 1 down 4 on the hundred digits)

Therefore, *** should be able to represent 32+24+16+8+1=81 numbers